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[tex]P(m)=\frac{n!}{(n-m)!m!}p^m(1-p)^{n-m} [/tex]

using Stirling's formula:

[tex]n!=n^n e^{-n} \sqrt{2\pi n} [/tex]

reduces to the normal distribution:

[tex]P(m)=\frac{1}{\sqrt{2 \pi n}} \frac{1}{\sqrt{p(1-p)}}

exp[-\frac{1}{2}\frac{(m-np)^2}{np(1-p)}]

[/tex]

Unfortunately, I keep on getting an extra term linear in m-np:

[tex] exp[\frac{m-np}{2n}\frac{2p-1}{(1-p)p}][/tex]

This term is zero if p=1/2, but I want to show that the binomial distribution reduces to the normal distribution for any probablity p.

Has anyone else had this problem, as I'm sure this derivation is fairly common?